So it sounds like if I'm making the syrup, I don't heat the sugar, correct?

I understand your confusion. The recipe is rather unclear as to how best to proceed. The directions to create the tincture are clear enough- adding the crushed berries to the neutral spirit and leaving it to rest is correct. However, I believe it will take more than a week to extract the flavors- and would probably opt to leave it go for at least two weeks, if not a full month. It will be, or should be a rather stout formulation at that point, I believe you would then make up the simple syrup, of equal parts water and sugar- and heat that until the sugar dissolves. When it has cooled to lukewarm, then you could mix in the tincture. That would store nicely in the fridge for a few weeks at best. You would not want to heat the alcohol based tincture, since the alcohol and flavor will evaporate. However, if you did not mix the tincture into the syrup- it will last indefinitely, so long as it is kept tightly capped. It won't spoil, unlike the syrup which can mold. You could flavor syrup with a few drops of the tincture as you wish, or simply add the drops into a drink you wish to flavor. Tasting a few drops will guide you as to how much to add.